Reduction of the Twisted Bilayer Graphene Chiral Hamiltonian into a $2\times2$ matrix operator and physical origin of flat-bands at magic angles

TL;DR

This paper simplifies the chiral Hamiltonian of twisted bilayer graphene into a 2x2 matrix form, revealing the physical origin of flat bands at magic angles and connecting them to floppy modes in flexible structures.

Contribution

It introduces a renormalized 2x2 matrix Hamiltonian that captures the physics of flat bands and their topological origin in twisted bilayer graphene.

Findings

Flat bands at magic angles are similar to floppy modes in flexible materials.

The renormalized Hamiltonian maps zero-mode regions into the ground state.

Flat-bands are associated with phase frustration and massive degeneracy.

Abstract

The chiral Hamiltonian for twisted graphene bilayers is written as a $2\times2$ matrix operator by a renormalization of the Hamiltonian that takes into account the particle-hole symmetry. This results in an effective Hamiltonian with an average field plus and effective non-Abelian gauge potential. The action of the proposed renormalization maps the zero-mode region into the ground state. Modes near zero energy have an antibonding nature in a triangular lattice. This leads to a phase-frustration effect associated with massive degeneration, and makes flat-bands modes similar to confined modes observed in other bipartite lattices. Suprisingly, the proposed Hamiltonian renormalization suggests that flat-bands at magic angles are akin to floppy-mode bands in flexible crystals or glasses, making an unexpected connection between rigidity topological theory and magic angle twisted two-dimensional heterostructures physics.